Of the three numbers, the ratio of the first and the second is `8 : 9` and that of the second and third is `3 : 4`. If the product of the first and third number is 2400, then the second number is :

Options

A. 45
B. 40
C. 30
D. 55

Answer

To solve the problem step by step, we will first define the variables based on the given ratios and then use the information provided to find the second number.

### Step-by-Step Solution:

1. **Define the Variables:**
Let the first number be \( A \), the second number be \( B \), and the third number be \( C \).
According to the problem, the ratios are given as:
\[ \frac{A}{B} = \frac{8}{9} \quad \text{and} \quad \frac{B}{C} = \frac{3}{4} \]

2. **Express the Numbers in Terms of a Common Variable:**
From the first ratio \( \frac{A}{B} = \frac{8}{9} \), we can express \( A \) and \( B \) in terms of a variable \( x \):
\[ A = 8x \quad \text{and} \quad B = 9x \]

From the second ratio \( \frac{B}{C} = \frac{3}{4} \), we can express \( B \) and \( C \) in terms of another variable \( y \):
\[ B = 3y \quad \text{and} \quad C = 4y \]

3. **Equate the Expressions for \( B \):**
Since both expressions represent \( B \), we can set them equal to each other:
\[ 9x = 3y \]
Solving for \( y \) in terms of \( x \):
\[ y = 3x \]

4. **Substitute \( y \) into the Expression for \( C \):**
Now substitute \( y = 3x \) into the expression for \( C \):
\[ C = 4y = 4(3x) = 12x \]

5. **Use the Product of \( A \) and \( C \):**
According to the problem, the product of the first and third numbers is given as:
\[ A \times C = 2400 \]
Substituting the expressions for \( A \) and \( C \):
\[ (8x)(12x) = 2400 \]
Simplifying this gives:
\[ 96x^2 = 2400 \]

6. **Solve for \( x^2 \):**
Divide both sides by 96:
\[ x^2 = \frac{2400}{96} = 25 \]
Taking the square root of both sides:
\[ x = 5 \]

7. **Find the Value of \( B \):**
Now that we have \( x \), we can find \( B \):
\[ B = 9x = 9 \times 5 = 45 \]

### Final Answer:
The second number \( B \) is \( 45 \).